Superlattice-Induced Insulating States and Valley-Protected Orbits in Twisted Bilayer Graphene

Y. Cao, J. Y. Luo, V. Fatemi, S. Fang, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero

Phys. Rev. Lett. 117, 116804 – Published 7 September 2016

Abstract

Twisted bilayer graphene (TBLG) is one of the simplest van der Waals heterostructures, yet it yields a complex electronic system with intricate interplay between moiré physics and interlayer hybridization effects. We report on electronic transport measurements of high mobility small angle TBLG devices showing clear evidence for insulating states at the superlattice band edges, with thermal activation gaps several times larger than theoretically predicted. Moreover, Shubnikov–de Haas oscillations and tight binding calculations reveal that the band structure consists of two intersecting Fermi contours whose crossing points are effectively unhybridized. We attribute this to exponentially suppressed interlayer hopping amplitudes for momentum transfers larger than the moiré wave vector.

Received 7 June 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.116804

Physics Subject Headings (PhySH)

Band gap Electrical conductivity Quantum Hall effect

Graphene Superlattices Van der Waals systems

First-principles calculations

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Y. Cao¹, J. Y. Luo¹, V. Fatemi¹, S. Fang², J. D. Sanchez-Yamagishi², K. Watanabe³, T. Taniguchi³, E. Kaxiras^2,4, and P. Jarillo-Herrero^1,*

¹Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
³National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan
⁴John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

^*pjarillo@mit.edu

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 117, Iss. 11 — 9 September 2016

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Schematic of TBLG and its superlattice unit cell. $λ_{SL} = {a / [2 \sin (θ / 2)]}$ ( $a$ is the lattice constant of graphene) is the moiré period and $A_{SL} = (\sqrt{3} / 2) λ_{SL}^{2}$ is the unit cell area. (b) The orange and blue hexagons denote the original Brillouin zones of graphene layer 1 and 2, respectively. In $k$ space, the band structure is folded into the MBZ which is defined by the mismatch between the hexagonal Brillouin zones of the two honeycomb lattices. (c) Illustration of the cross section of our device. (d) Optical image of $θ \approx 1.8 °$ device $S 1$ . The hall bar in the dashed rectangular region is completely free of bubbles and ridges.
Reuse & Permissions
Figure 2
(a) Comparison of the conductivity of a large angle TBLG device $S 0$ and a small angle device $S 1$ . The vertical bars around $n = \pm 7.5 \times 1 0^{12} {cm}^{- 2}$ indicate the insulating states in device $S 1$ . (b) Tight-binding band structure of TBLG with $θ = 1.8 °$ . Dashed lines denote the monolayer graphene dispersion with Fermi velocity $v_{F} = 1 \times 1 0^{6} m s^{- 1}$ . The color of the bands denotes the valley polarization: $K$ (orange), $K^{'}$ (navy blue), and valley degenerate (purple). The arrows indicate the direct band gaps at $Γ_{s}$ . (c) Temperature dependent conductivity of device $S 1$ . (d) Arrhenius plot of the conductivity of the insulating states [indicated by dashed lines in (c)]. Blue and red denote the electron and hole side insulating states, respectively. Thermal activation gaps of $\sim 50$ and $\sim 60 meV$ are estimated from the slope for the electron-side and hole-side insulating states, respectively.
Reuse & Permissions
Figure 3
(a) Longitudinal resistivity and (b) Hall conductivity in units of $σ_{0} = e^{2} / h$ versus the total density and the magnetic field. Measurements are taken at $T = 40 mK$ . (c) Reconstructed Landau level structure from the plateau values. The central Landau fan emanating from the Dirac point at zero density has an eightfold degenerate half-integer quantum Hall sequence, while the Landau fans originating from the superlattice gaps have a fourfold degenerate massive parabolic quantum Hall sequence.
Reuse & Permissions
Figure 4
(a) Cyclotron masses and (b) oscillation frequencies extracted from SdH measurements. The red curve is the numerically calculated cyclotron mass (normalized by a factor of 0.5) and the black dashed curve is the effective mass if the interlayer interaction is ignored. Lines colored pink, blue, and green denote the expected slope for the outer star orbit, triangular orbits, and inner hexagon orbit shown in (d) and (e). (c)–(e) Fermi contours at densities shown as arrows positioned below the density axis in (b). Orange orbits are $K$ polarized, and blue orbits are $K^{'}$ polarized. (f) 3D illustration of the low-energy band structure. The two sets of bands are valley polarized in the original $K$ , $K^{'}$ valleys of the constituent layers. For example, the $K$ subbands result from the hybridization of $K^{(1)}$ and $K^{(2)}$ Dirac cones. The same applies for the $K^{'}$ subbands.
Reuse & Permissions

Physical Review Letters